Stats Exam 2

trimmed mean

deleting percentage of smallest and largest values from data set and computing mean of the rest

bimodel

data set with 2 modes

multimodal

data set with more than 2 modes

weighted mean

(weight of obs x value of obs) / weight of obs

geometric mean definition

used in analyzing growth rates in financial data; applied anytime want to determine mean rate of change over successive periods

geometric mean formula

xg = nth root of [(x1)(x2)…(xn)]

percentile definition

at least p% of items take on this value or less, at least (100-p)% of items take on this value or more

location of pth percentile

Lp = (p/100)(n-1)

1st quartile

25th percentile

2nd quartile

50th percentile

3rd quartile

75th percentile

measures of variability

range, interquartile range, variance, standard deviation

interquartile range formula

IQR = Q3 - Q1

variance definition

difference between xi and the mean

sample variance formula

s² = (sum of [xi-x]²) / (n-1)

population variance formula

S² = (sum of [xi-u]²) / (N)

standard deviation definition

measures the dispersion of a dataset relative to its mean

standard deviation formula

square root of the variance (s for sample, omega for population)

coefficient of variation definition

indicates how large the standard deviation is in relation to the mean

coefficient of variation formula

CV = (population standard deviation) / (population mean)

left skewness

skewness is negative, mean < median, tail to the left

right skewness

skewness is positive, mean > median, tail is to the right

highly skewed right

skewness is often above 1.0

z-scores definition

number of standard deviations a data value is from the mean

z-score formula

zi = (xi-x) / (s)

negative z-score

data value < sample mean

positive z-score

data value > sample mean

Chebyshev’s Theorem

at least 1 - (1/z²) of items will be within z standard deviations of the mean

z=2

at least 75% data points

z=3

at least 89% data points

z=4

at least 94% data points

empirical rule definition

when data approximates a bell-shaped distribution, can determine % of values within specified number of standard deviations

empirical rule ± 1 standard deviation

68%

empirical rule ± 2 standard deviations

95%

empirical rule ± 3 standard deviations

99.7%

outlier characteristics

± 3 standard deviations away, incorrectly recorded, included, or just unusual

boxplot requirements

smallest value, first quartile, median, third quartile, largest value

lower limit formula

Q1 = 1.5(IQR) below Q1

upper limit

Q3 = 1.5(IQR) above Q3

covariance definition

linear association between two variables

sample covariance formula

sxy = (sum of [(xi-x)(yi-y)]) / (n-1)

population covariance formula

sigmaxy = (sum of [(xi-Mx)(yi-Yy)]) / (N)

correlation coefficient definition

linear association, not necessarily causation, can take on values between -1 and 1

sample correlation coefficient formula

rxy = (sxy) / [(sx)(sy)]

population correlation coefficient formula

Pxy = (sigmaxy) / [(sigmax)(sigmay)]

strong negative correlation

-1

strong positive correlation

1

data dashboard characteristics

graphical and numerical

drilling down (data dashboards)

functionality in interactive dashboards that allow user to access information at a detailed level

probability definition

likelihood an event will occur, scale from 0-1

stats experiment

generates a well-defined outcome

sample space

set of all experimental outcomes

sample point

experimental outcome

coin toss experiment

the actual toss

coin toss sample space

heads, tails

counting rule for multiple step experiment definition

sequence of k steps in which there are n1 possible results for first step, n2 for second step…

total number of experimental outcomes formula

(n1)(n2)k(nk)

combinations formula

CNn = (N!) / (n!)(N-n)!

permutations formula

PNn = (n!) [(N!) / (N-n)!]

event

collection of sample points

probability

sum of probabilities of sample points in the event

complement of event definition

event consisting of all sample points NOT in A; P(A) = P(Ac)

Union of two events definition

event containing all sample points IN A OR B OR BOTH; A U B

intersection of two events definition

set of all sample points in BOTH A AND B

mutually exclusive events definition

events have no sample points in common, when one event occurs the other can’t occur

conditional probability definition

probability of event given that another event has occurred

(independent) P(A and B)=

P(A)P(B)

(dependent) P(A and B)=

P(A)P(B|A)

independent events definition

if probability of A is not changed by the existence of B

if one mutually exclusive event is known to occur,

the other event’s probability is 0

2 events that are not mutually exclusive,

might or might not be independent

Bayes’ Theorem

prior probability, likelihood of the evidence, and posterior probability

Bayes’ Theorem formula

P(A|B) = [P(B|A)P(A)] / P(B)

probability distribution definition

describes how probabilities are distributed over values of random variables

discrete uniform probability distribution

f(x) = 1/n

discrete uniform expected value

E(x) = xf(x)

discrete uniform variance

(x-u)²f(x)

bivariate distribution definition

2 random variables being looked at, interested in relationship between them

binomial probability distribution characteristics

experiment with n identical trials, 2 outcomes success and failure on each, prob of success is p, does not change from trial to trial, trials are independent

binomial prob dist formula

f(x) = (n!) / [x!(n-x)!] p^x (1-p)^(n-x)

binomial prob dist variance formula

var(x) = np(1-p)

poisson probability distribution characteristics

prob of occurrence is same for any 2 intervals of equal length, occurrence or non in interval is independent of occurrence in any other interval, number of occurrences per interval

poisson formula

f(x) = [(M^x)(e^-M)] / x!

hypergeometric prob dist

trials not independent, prob of success does change from trial to trial

uniform prob dist expected value

E(x) = (a+b)/n

uniform prob dist variance

(b-a) / n

normal prob dist characteristics

skewness is 0, mean median mode same

z-score formula

z = (x-M) / standard dev

exponential prob dist characteristics

time takes to complete a task, mean and standard dev are same, length of interval between occurrences

exponential prob formula

p(x-x0) = 1+e^(-x0/M)